Probability Distribution of Discrete Random Variables

An experiment where the result is not predetermined, and the outcome is uncertain.

A variable that takes on numerical values based on the outcomes of a random experiment.

A variable that can take on a finite number of values or a countably infinite number of values.

A variable that can take on any value within a given range.

A table or graph that shows the probability of each possible outcome of a discrete random variable.

A discrete random variable where the probability of success in each trial is constant and trials are independent.

A discrete random variable that represents the number of events occurring in a fixed period of time or place when events are independent and occur at a constant average rate.

The probability of a specific outcome of a discrete random variable must be between 0 and 1 (inclusive), and the sum of all probabilities for all possible outcomes must equal 1.

A random variable is a variable whose value is a numerical outcome of a random phenomenon.

A discrete random variable is one whose value can only take on specific, distinct values.

A probability distribution lists all possible values of a discrete random variable and their corresponding probabilities.

The expected value is the average value you would expect to obtain if you repeated an experiment many times.

The variance measures the spread of the probability distribution, or how far the values are, on average, from the mean.

The standard deviation is the square root of the variance.

The probability of an event is the likelihood that the event will occur.

The mean of a discrete variable is calculated as the sum of each value multiplied by its probability.

The number of events occurring in a fixed period of time or place when events are independent and occur at a constant average rate.

The probability of each possible outcome must be between 0 and 1, and the sum of all probabilities for all possible outcomes must equal 1.

Events occurring independently at a constant average rate.

The Poisson Distribution describes the probability of a specific number of events occurring within a fixed interval of time or space, given that these events happen independently and at a constant rate.

The average number of events that occur within the specified interval.

The formula calculates the probability of getting 'x' events in the defined interval, where λ is the average rate, 'e' is Euler's number (approximately 2.71828), and 'x!' represents the factorial of 'x'.

The Poisson Distribution is relevant when dealing with independent events occurring at a constant average rate, such as the number of customers arriving at a store, the number of defects in a manufacturing process, the number of meteorites hitting Earth, and many others.

The average number of events per unit or interval, represented by λ, is also the variance of the Poisson Distribution.

The shape of the Poisson Distribution's probability graph depends on λ. As λ increases, the distribution becomes more symmetrical.

The Poisson Distribution assumes events are independent, meaning the occurrence of one event doesn't affect the probability of another event.

The Poisson Distribution is a useful tool for modeling and analyzing rare events happening randomly over time or space.

The standard deviation of a discrete random variable is a measure of its spread or variability. It is calculated as the square root of the variance, which measures how far, on average, the values of the variable deviate from its mean.

The expected value (E(X)) of a discrete random variable X is the average of all possible values of X, weighted by their respective probabilities. It represents the long-run average value of the variable.

The variance (Var(X)) of a discrete random variable X is a measure of its spread or variability. It is calculated as the average squared deviation of the values of X from its mean.

Chebyshev's rule states that for any discrete random variable, at least 1 - 1/k^2 of the data values will fall within k standard deviations of the mean, where k is any number greater than 1.

The empirical rule is a statistical rule that applies to data that follows a normal distribution. It states that approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and nearly 100% falls within three standard deviations.

A Bernoulli trial is a random experiment with only two possible outcomes: success or failure. The probability of success (p) is constant and the trials are independent.

A binomial experiment is a sequence of Bernoulli trials with a fixed number of trials, where each trial is independent and has the same probability of success.

A binomial random variable is a discrete variable that counts the number of successes in a fixed number of Bernoulli trials.

The average number of events expected to occur in a given time period or place.

The probability of observing exactly x events within the given time period or place.

The probability of observing at least x events within the given time period or place.

The probability of observing fewer than x events within the given time period or place.

The probability of getting exactly x successes in n trials (n ≥ x ), with probability of success p in each trial.

The probability of getting at least x successes in n trials.